OverlapLowerBoundOS
plain-language theorem explainer
OverlapLowerBoundOS encodes the predicate that a real parameter β satisfies 0 < β ≤ 1 for any transfer kernel K. Lattice gauge theorists constructing Osterwalder-Schrader positivity surrogates cite this predicate when building spectral guards for transfer operators. The definition is a direct conjunction of two inequalities with no lemmas or tactics applied.
Claim. For a transfer kernel $K$ and real number $β$, the overlap lower bound condition holds precisely when $0 < β ≤ 1$.
background
The Kernel structure in YM.OS is a transfer operator T that maps continuous linear functionals from complex observables over LatticeMeasure to themselves. This supports positivity conditions in the Yang-Mills setting of the Recognition Science mirror. The module defines related objects including LatticeMeasure and TransferOperator, with the overlap predicate serving as a uniform lower bound on β to ensure contraction properties compatible with Dobrushin-type estimates.
proof idea
The declaration is a direct definition that states the conjunction 0 < β ∧ β ≤ 1. No upstream lemmas are invoked; the body is the predicate itself.
why it matters
This predicate is the building block for OSPositivity, which asserts existence of a kernel K and β satisfying the bound, and for the default lemma that instantiates it with β = 1. It supplies the spectral positivity guard in the YM.OS module and connects to the broader transfer-operator treatment of mass-gap properties. In the Recognition framework it supports the eight-tick octave and phi-ladder constructions by ensuring uniform overlap in the discrete-to-continuum passage.
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